Euler Product for the Zeta Function

ζ(s) = 1+

1

s

2

+

1

s

3

+

1

s

4

+⋯

ζ(s)1-

1

s

2

= 1+

1

s

3

+

1

s

5

+

1

s

7

+⋯

ζ(s)1-

1

s

2

1-

1

s

3

= 1+

1

s

5

+

1

s

7

+

1

s

11

+⋯

The Riemann zeta function can be represented by the Dirichlet series:

ζ(s)=

∞

∑

n=1

1

s

n

=1+

1

s

2

+

1

s

3

+

1

s

4

+⋯

,

which converges for real

s

> 1.

Euler derived the remarkable product formula:

ζ(s)

∞

∏

n=1

1-

1

p

s

(n)

=1

, where

p(n)

represents the

th

n

prime:

p(1)=2

,

p(2)=3

,

p(3)=5

, ….

This represents a very suggestive relationship between prime numbers and the Riemann zeta function. It has been called "The Golden Key" [J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, New York: Penguin, 2004].

On a more mundane level, the author has found this derivation to be an excellent student exercise in manipulating infinite sums and products. This Demonstration enables the stepwise evaluation of the product as the number of prime factors is increased. The deviation of each partial result from 1 is shown on a log-log plot as a function of

s

and the number of factors. The difference rapidly approaches a large negative power of 10.